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Colouring a Torus

If a map is drawn on a torus,colours are required to colour each region or country, with countries adjacent to each other - having a common border, being coloured differently. Note that countries meeting at a single point are not considered to be adjacent and may be coloured the same.

To show at least seven colours are required, we only need to draw a map on a torus that requires seven colours. One is shown below. More elaborate maps may require more colours.

The Euler characteristic of the torus isFrom this link, we obtain the number of colours required,is at most seven. Hence